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May 2022 Local elliptic law
Johannes Alt, Torben Krüger
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Bernoulli 28(2): 886-909 (May 2022). DOI: 10.3150/21-BEJ1370

Abstract

The empirical eigenvalue distribution of the elliptic random matrix ensemble tends to the uniform measure on an ellipse in the complex plane as its dimension tends to infinity. We show this convergence on all mesoscopic scales slightly above the typical eigenvalue spacing in the bulk spectrum with an optimal convergence rate. As a corollary we obtain complete delocalisation for the corresponding eigenvectors in any basis.

Funding Statement

The first author gratefully acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 895698, from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715539 RandMat) and from the Swiss National Science Foundation through the NCCR SwissMAP grant.
The second author gratefully acknowledges financial support from Novo Nordisk Fonden Project Grant 0064428 & VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and Young Investigator Award (Grant No. 29369).

Citation

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Johannes Alt. Torben Krüger. "Local elliptic law." Bernoulli 28 (2) 886 - 909, May 2022. https://doi.org/10.3150/21-BEJ1370

Information

Received: 1 February 2021; Revised: 1 May 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

Digital Object Identifier: 10.3150/21-BEJ1370

Keywords: eigenvector delocalisation , elliptic ensemble , Local law , matrix Dyson equation

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 2 • May 2022
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